Advertisements
Advertisements
प्रश्न
Find the unit vectors perpendicular to each of the vectors `vec"a" + vec"b"` and `vec"a" - vec"b"`, where `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`
Advertisements
उत्तर
Th given vectors are `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`
`(vec"a" + vec"b") xx (vec"a" - vec"b") = vec"a" xx vec"a" - vec"a" xx vec"b" + vec"b" xx vec"a" - vec"b" xx vec"b"`
= `0 - vec"a" xx vec"b"- vec"a" xx vec"b" - 0`
= `- 2 vec"a" xx vec"b"`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(1, 1, 1),(1, 2, 3)|`
= `hat"i"(3 - 2) - hat"j"(3 - 1) + hat"k"(2 - 1)`
`vec"a" xx vec"b" = hat"i" - 2hat"j" + hat"k"`
`|vec"a" xx vec"b"| = |hat"i" - 2hat"j" + hat"k"|`
= `sqrt(1^2 + (-2)^2 + 1^2)`
= `sqrt(1 + 4 + 1)`
= `sqrt(6)`
The unit vector perpendicular to `vec"a" + vec"b"` and
`vec"a" - vec"b"` is = `+- ((vec"a" + vec"b") xx (vec"a" - vec"b"))/|(vec"a" + vec"b") xx (vec"a" - vec"b")|`
= `+- (-2(vec"a" xx vec"b"))/(|-2(vec"a" xx vec"b")|)`
= `+- (-2(hat"i" - 2hat"j" + hat"k"))/(2 xx sqrt(6))`
= `+- ((-hat"i" + 2hat"j" - hat"k"))/sqrt(6)`
APPEARS IN
संबंधित प्रश्न
If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that
`sin theta/2 = 1/2|vec"a" - vec"b"|`
If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that
`tan theta/2 = |vec"a" - vec"b"|/|vec"a" + vec"b"|`
Find `vec"a"*vec"b"` when `vec"a" = hat"i" - 2hat"j" + hat"k"` and `vec"b" = 3hat"i" - 4hat"j" - 2hat"k"`
Find the value λ for which the vectors `vec"a"` and `vec"b"` are perpendicular, where `vec"a" = 2hat"i" + lambdahat"j" + hat"k"` and `vec"b" = hat"i" - 2hat"j" + 3hat"k"`
Find the value λ for which the vectors `vec"a"` and `vec"b"` are perpendicular, where `vec"a" = 2hat"i" + 4hat"j" - hat"k"` and `vec"b" = 3hat"i" - 2hat"j" + lambdahat"k"`
If `vec"a"` and `vec"b"` are two vectors such that `|vec"a"| = 10, |vec"b"| = 15` and `vec"a"*vec"b" = 75sqrt(2)`, find the angle between `vec"a"` and `vec"b"`
Find the angle between the vectors
`hat"i" - hat"j"` and `hat"j" - hat"k"`
Show that the vectors `-hat"i" - 2hat"j" - 6hat"k", 2hat"i" - hat"j" + hat"k"` and find `-hat"i" + 3hat"j" + 5hat"k"` form a right angled triangle
Let `vec"a", vec"b", vec"c"` be three vectors such that `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 5` and each one of them being perpendicular to the sum of the other two, find `|vec"a" + vec"b" + vec"c"|`
Find λ, when the projection of `vec"a" = lambdahat"i" + hat"j" + 4hat"k"` on `vec"b" = 2hat"i" + 6hat"j" + 3hat"k"` is 4 units
Find the magnitude of `vec"a" xx vec"b"` if `vec"a" = 2hat"i" + hat"j" + 3hat"k"` and `vec"b" = 3hat"i" + 5hat"j" - 2hat"k"`
Show that `vec"a" xx (vec"b" + vec"c") + vec"b" xx (vec"c" + vec"a") + vec"c" xx (vec"a" + vec"b") = vec0`
Find the vectors of magnitude `10sqrt(3)` that are perpendicular to the plane which contains `hat"i" + 2hat"j" + hat"k"` and `hat"i" + 3hat"j" + 4hat"k"`
Find the area of the parallelogram whose two adjacent sides are determined by the vectors `hat"i" + 2hat"j" + 3hat"k"` and `3hat"i" - 2hat"j" + hat"k"`
Choose the correct alternative:
A vector `vec"OP"` makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between `vec"OP"` and the z-axis is
Choose the correct alternative:
If `vec"a"` and `vec"b"` having same magnitude and angle between them is 60° and their scalar product `1/2` is then `|vec"a"|` is
Choose the correct alternative:
If `|vec"a"| = 13, |vec"b"| = 5` and `vec"a" * vec"b"` = 60° then `|vec"a" xx vec"b"|` is
Choose the correct alternative:
If `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = 2hat"i" + xhat"j" + hat"k", vec"c" = hat"i" - hat"j" + 4hat"k"` and `vec"a" * (vec"b" xx vec"c")` = 70, then x is equal to
